When is $\max_x\{f(g(x))\} = f(\max_x\{g(x)\})$?

Question

What is the requirements on $f$ and $g$ in order for $\max_x\{f(g(x))\} = f(\max_x\{g(x)\})$ to be correct?

Equivalently, when is $\min_x\{f(g(x))\} = f(\min_x\{g(x)\})$ ?

Any reference for describing this is welcome.

Answer 1

A sufficient condition for $\max_x\{f(g(x))\} = f(\max_x\{g(x)\})$ is that $f$ is a monotone increasing function on the range of $g$. This is also sufficient condition for $\min_x\{f(g(x))\} = f(\min_x\{g(x)\})$ and the reason is that monotone increasing functions are order preserving.

To clarify, a function is monotone increasing if $x \leq y$ implies $f(x) \leq f(y)$. Thus also $g(x) \leq g(y)$ implies $f(g(x)) \leq f(g(y))$. Consequently, each minimizer (resp. maximizer) of $g(x)$ is also a minimizer (resp. maximizer) of $f(g(x))$.

In contrast, monotone decreasing functions are order reversing, and you would obtain relations such as $\max_x\{f(g(x))\} = f(\min_x\{g(x)\})$.